Subgroups — C₂×#20295581 #28376321

Order 13952.

1168 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295581	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16812699	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812690	C₂	1	2	yes	maximal, complemented
4	C₂×#16812679	C₂	1	2	yes	maximal
5	C₂×#16812675	C₂	1	2	yes	maximal
6	C₂×Q₈×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
7	C₂×#16812664	C₂	1	2	yes	maximal, complemented
8	C₂×#16812657	C₂	1	2	yes	maximal
9	C₂×C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
12	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
15	#16812699	C₂²	1	8	yes	complemented
16	#16812690	C₂²	1	16	yes	complemented
17	#16812679	C₂²	1	16	yes	—
18	#16812675	C₂²	1	16	yes	—
19	Q₈×Dic₁₀₉	C₂²	1	16	yes	complemented
20	#16812664	C₂²	1	16	yes	complemented
21	#16812657	C₂²	1	16	yes	—
22	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
25	C₂×#13182008	C₂²	1	4	yes	—
26	C₂×#13182006	C₂²	1	4	yes	—
27	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
31	C₄²×D₁₀₉	C₂²	1	8	yes	—
32	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
33	C₄⋊Q₈×C₁₀₉	C₂²	1	8	yes	complemented
34	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
38	C₂×Dic₂₁₈	—	2	8	no	—
39	C₂×Dic₂₁₈	C₂³	1	8	yes	—
40	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
43	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₁₀₉	—	2	4	no	—
45	C₂×C₄×D₁₀₉	—	2	4	no	—
46	C₂×C₄×D₁₀₉	Q₈	1	8	yes	complemented
47	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
48	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
49	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
50	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
51	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
52	C₄⋊Dic₁₀₉	C₂³	1	16	yes	—
53	#13182008	C₂³	1	16	yes	—
54	#13182006	C₂³	1	16	yes	—
55	C₄×Dic₁₀₉	C₂³	1	4	yes	—
56	C₄×Dic₁₀₉	C₂³	1	4	yes	—
57	C₄×Dic₁₀₉	C₂³	1	4	yes	—
58	C₄×Dic₁₀₉	C₂³	1	16	yes	—
59	C₂×Q₈×C₁₀₉	—	2	8	no	—
60	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
61	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
62	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
66	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
67	C₄²×C₁₀₉	C₂³	1	4	yes	—
68	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	C₂²×D₁₀₉	—	2	2	no	—
70	C₂²×D₁₀₉	—	2	2	no	—
71	C₂²×D₁₀₉	C₂×Q₈	1	4	yes	—
72	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
73	Q₈×C₁₀₉	—	2	16	no	—
74	C₂×C₄×C₁₀₉	—	2	8	no	—
75	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₁₀₉	#41	1	4	yes	—
77	C₂×C₄×C₁₀₉	#41	1	4	yes	—
78	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
79	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
80	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
81	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
82	C₂×Dic₁₀₉	—	2	2	no	—
83	C₂×Dic₁₀₉	—	2	2	no	—
84	C₂×Dic₁₀₉	—	2	8	no	—
85	C₂×Dic₁₀₉	C₂×Q₈	1	4	yes	—
86	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
87	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
88	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
89	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
90	C₄×D₁₀₉	—	2	8	no	—
91	C₄×D₁₀₉	—	2	8	no	—
92	C₄×D₁₀₉	C₂×Q₈	1	16	yes	complemented
93	Dic₂₁₈	—	2	16	no	—
94	C₂²×C₁₀₉	C₂²×Q₈	1	2	yes	—
95	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
96	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
97	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
98	C₂×D₁₀₉	—	2	8	no	—
99	C₂×D₁₀₉	—	2	2	no	—
100	C₂×D₁₀₉	—	2	2	no	—
101	C₂×D₁₀₉	C₂²×Q₈	1	4	yes	—
102	C₄×C₁₀₉	—	2	8	no	—
103	C₄×C₁₀₉	C₂²×Q₈	1	4	yes	—
104	C₄×C₁₀₉	C₂×#41	1	4	yes	—
105	C₄×C₁₀₉	C₂×#41	1	4	yes	—
106	Dic₁₀₉	—	2	8	no	—
107	Dic₁₀₉	—	2	2	no	—
108	Dic₁₀₉	—	2	2	no	—
109	Dic₁₀₉	C₂²×Q₈	1	4	yes	—
110	C₂×C₁₀₉	#533	1	4	yes	complemented
111	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
112	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
113	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
114	D₁₀₉	—	2	4	no	—
115	C₂×#533	—	109	1	no	maximal, Sylow
116	C₁₀₉	C₂×#533	1	1	yes	characteristic, complemented, Sylow
117	#533	—	109	16	no	—
118	C₂×C₄⋊Q₈	—	109	1	no	—
119	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
120	C₂×#125	—	109	2	no	—
121	C₂×C₂²⋊Q₈	—	109	2	no	—
122	C₂×C₂²⋊Q₈	—	109	2	no	—
123	C₂×C₄×Q₈	—	109	2	no	—
124	C₂×C₄×Q₈	—	109	2	no	—
125	C₂×#117	—	109	2	no	—
126	C₂²×C₄²	—	109	1	no	—
127	C₂²×Q₈	—	109	2	no	—
128	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
129	C₂³×C₄	—	109	1	no	—
130	C₂³×C₄	—	109	1	no	—
131	C₂³×C₄	—	109	1	no	—
132	C₄⋊Q₈	—	109	8	no	—
133	C₄⋊Q₈	C₂×D₁₀₉	1	8	yes	complemented
134	#125	—	109	16	no	—
135	C₂²⋊Q₈	—	109	16	no	—
136	C₂²⋊Q₈	—	109	16	no	—
137	C₄×Q₈	—	109	16	no	—
138	C₄×Q₈	—	109	16	no	—
139	#117	—	109	16	no	—
140	C₂×C₄⋊C₄	—	109	2	no	—
141	C₂×C₄⋊C₄	—	109	4	no	—
142	C₂×C₄⋊C₄	—	109	4	no	—
143	C₂×C₄⋊C₄	—	109	2	no	—
144	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
145	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
146	C₂×C₂²⋊C₄	—	109	4	no	—
147	C₂×C₄²	—	109	4	no	—
148	C₂×C₄²	—	109	1	no	—
149	C₂×C₄²	—	109	1	no	—
150	C₂×C₄²	—	109	1	no	—
151	C₂×C₄²	—	109	8	no	—
152	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
153	C₂⁴	—	109	1	no	—
154	C₂×Q₈	—	2	8	no	—
155	C₂×Q₈	—	109	8	no	—
156	C₂×Q₈	—	218	8	no	—
157	C₂×Q₈	C₂²×D₁₀₉	1	8	yes	—
158	C₂²×C₄	—	109	1	no	—
159	C₂²×C₄	—	109	8	no	—
160	C₂²×C₄	—	109	4	no	—
161	C₂²×C₄	—	109	1	no	—
162	C₂²×C₄	—	109	1	no	—
163	C₂²×C₄	—	109	4	no	—
164	C₂²×C₄	—	109	4	no	—
165	C₂²×C₄	—	109	4	no	—
166	C₂²×C₄	—	218	4	no	—
167	C₂²×C₄	—	218	4	no	—
168	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
169	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
172	C₄⋊C₄	—	109	16	no	—
173	C₄⋊C₄	—	109	8	no	—
174	C₄⋊C₄	—	109	16	no	—
175	C₄⋊C₄	—	109	8	no	—
176	C₄⋊C₄	C₂²×D₁₀₉	1	8	yes	—
177	C₄⋊C₄	C₂²×D₁₀₉	1	8	yes	—
178	C₂²⋊C₄	—	109	16	no	—
179	C₄²	—	109	4	no	—
180	C₄²	—	109	4	no	—
181	C₄²	—	109	4	no	—
182	C₄²	—	109	16	no	—
183	C₄²	C₂²×D₁₀₉	1	4	yes	—
184	C₂³	—	109	4	no	—
185	C₂³	—	109	2	no	—
186	C₂³	—	218	2	no	—
187	C₂³	—	218	2	no	—
188	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	218	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	109	4	no	—
193	C₂×C₄	—	109	2	no	—
194	C₂×C₄	—	109	16	no	—
195	C₂×C₄	—	109	2	no	—
196	C₂×C₄	—	109	2	no	—
197	C₂×C₄	—	109	8	no	—
198	C₂×C₄	—	218	2	no	—
199	C₂×C₄	—	218	2	no	—
200	C₂×C₄	—	218	8	no	—
201	C₂×C₄	—	218	8	no	—
202	C₂×C₄	—	218	8	no	—
203	C₂×C₄	Q₈×D₁₀₉	1	4	yes	—
204	C₂×C₄	#13182028	1	4	yes	—
205	C₂×C₄	#13182030	1	4	yes	—
206	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
207	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
208	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
209	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
210	C₂²	—	109	4	no	—
211	C₂²	—	218	8	no	—
212	C₂²	—	218	2	no	—
213	C₂²	—	218	2	no	—
214	C₂²	C₂×Q₈×D₁₀₉	1	2	yes	—
215	C₂²	C₂×#13182028	1	2	yes	—
216	C₂²	C₂×#13182030	1	2	yes	—
217	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
218	C₄	—	2	8	no	—
219	C₄	—	109	4	no	—
220	C₄	—	218	2	no	—
221	C₄	—	218	2	no	—
222	C₄	—	218	8	no	—
223	C₄	C₂×Q₈×D₁₀₉	1	4	yes	—
224	C₄	C₂×#13182028	1	4	yes	—
225	C₄	C₂×#13182030	1	4	yes	—
226	C₂	—	218	4	no	—
227	C₂	#20295581	1	4	yes	direct product factor
228	C₂	C₂²×Q₈×D₁₀₉	1	1	yes	characteristic
229	C₂	C₂²×#13182028	1	1	yes	characteristic
230	C₂	C₂²×#13182030	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂³×D₁₀₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#533
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20295581	C₂	direct (N × H)
2. C₂×#16812699	C₂	semidirect
3. C₂×#16812690	C₂	semidirect
6. C₂×Q₈×Dic₁₀₉	C₂	semidirect
7. C₂×#16812664	C₂	semidirect
10. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
15. #16812699	C₂²	semidirect
16. #16812690	C₂²	semidirect
19. Q₈×Dic₁₀₉	C₂²	semidirect
20. #16812664	C₂²	semidirect
33. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
46. C₂×C₄×D₁₀₉	Q₈	semidirect
92. C₄×D₁₀₉	C₂×Q₈	semidirect
110. C₂×C₁₀₉	#533	semidirect
116. C₁₀₉	C₂×#533	semidirect
119. C₂×C₄⋊Q₈	D₁₀₉	semidirect
133. C₄⋊Q₈	C₂×D₁₀₉	semidirect
227. C₂	#20295581	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295581	2
C₂×#16812699	2
C₂×#16812690	2
C₂×#16812679	2
C₂×#16812675	2
C₂×Q₈×Dic₁₀₉	2
C₂×#16812664	2
C₂×#16812657	2
C₂×C₄²×D₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#533	109

Subgroups — C2×#20295581 #28376321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295581 #28376321